Fluid friction in partially filled circular conduits

Publisher’s version / Version de l'éditeur:

Transactions, Engineering Institute of Canada, 1, pp. 39-43, 1957-11-01

READ THESE TERMS AND CONDITIONS CAREFULLY BEFORE USING THIS WEBSITE. https://nrc-publications.canada.ca/eng/copyright

Vous avez des questions? Nous pouvons vous aider. Pour communiquer directement avec un auteur, consultez la

première page de la revue dans laquelle son article a été publié afin de trouver ses coordonnées. Si vous n’arrivez pas à les repérer, communiquez avec nous à [email protected].

Questions? Contact the NRC Publications Archive team at

[email protected]. If you wish to email the authors directly, please see the first page of the publication for their contact information.

NRC Publications Archive

Archives des publications du CNRC

This publication could be one of several versions: author’s original, accepted manuscript or the publisher’s version. / La version de cette publication peut être l’une des suivantes : la version prépublication de l’auteur, la version acceptée du manuscrit ou la version de l’éditeur.

Access and use of this website and the material on it are subject to the Terms and Conditions set forth at

Fluid friction in partially filled circular conduits

Stephenson, D. G.

https://publications-cnrc.canada.ca/fra/droits

L’accès à ce site Web et l’utilisation de son contenu sont assujettis aux conditions présentées dans le site

LISEZ CES CONDITIONS ATTENTIVEMENT AVANT D’UTILISER CE SITE WEB.

NRC Publications Record / Notice d'Archives des publications de CNRC:

https://nrc-publications.canada.ca/eng/view/object/?id=9320430c-0322-4fad-9d6b-d5fa6ae9995d

https://publications-cnrc.canada.ca/fra/voir/objet/?id=9320430c-0322-4fad-9d6b-d5fa6ae9995d

FLUID FRICTION IN PARTIALLY

FILLED CIRCULAR CONDUITS

by

D.

G.

Stephenson

REPRINTED FROM

TRANSACTIONS, ENGINEERING INSTITUTE OF CANADA NO. 1, SEPTEMBER, 1957, p. 39-43

RESEARCH PAPER No.

44,

O F T H E DIVISION O F BUILDING RESEARCH

thc vertical m:mo manometer rcspccti~ p = viscosity of tlic : ~ i r i

This paper r1r:~ls only with straight runs w h ~ r c thc entrance cffoets are negligible.

THE APPARATUS

f a n empty conduit with the same

sectioli.

P = the mean absolute static pressure in the test were brought together nr\rrre the same. As sI1on.n in all differential pressure measurements were made I 400-mm. water n~icromanometcr and the gauge pre

flow metcring orifice. were measured on a 100-in. nianometer with carbon

AP = pressure drop due to friction in L: length

L.

V . p . d V . p . d ,

these measurements is shown in Fig. 2. With this gement water could be drawn u p into the conduit t h e flask: the height t o which t h e water rose was ured with t h e manometer a n d t h e weight of water in

srction was so small t h a t thc>re was no significant differel

MODIFIED UNION

100" MANOMETER

Fig. 1. Arrangemen1 of the apparatus. ICach of the nurnbcred prcssure tappings was connected to one of the points on the pressurc: switcllitlg panel.

TESTS Measurement of Conduit Diameters

With a length of conduit suspcndccl vertically as in Fig. 2 the pressure inside the conduit x a s lowrrrd in steps of S in. of water. Each time the prc,ssurc was reduced thc vertical position of t h e conduit was adjusted t o keep the lower end 2 in. helow the surface of the water a n d then tlic weight of t h e beaker was recorded along wit11 t h e rcnding of t h e vertical manornetcr. D a t a were recorded only nrhen watcr was rising in a d r y concluit. T h e average cross-sectional arca for each S-in. segment of the pipe could be crtlculatccl from t h e change in weight of the beaker \vl1c11 the water filled t h a t segment. The csprcssion for t h e inran cross- sectional arca is

It

is not necessary t o know

ah,,

a n d p,, to a high accuracy since the protluct ah,,

.

p,, can be rcplacccl by t h e equal product ~ h , , , . p,,,. Tllc value of ah,, could be measured directly, a n d p,,, was chcckrcl Iwfore and after t h e measurements on each length of collduit by connecting the 100-in. vertical manometcr in p:1r:lllcl with a micro- mnnonletcr a n d applying a l)rc3ssurc,. Then

X .

P7,l = Y

.

Pb . . . (2) T h e ~ n i c r o n l a n o ~ n c t c r containccl tlistilletl nntvr so tli:~t pc,

could bc accurately dctcrminecl I,y measuring t h r temper:^- ture of t h e ~vatclr in tllc micron~:lnomctc~~.

bctwccn tlle root mean s q u : ~ r r tiiamctcr

'VG

and the - root mean fifth ' V Z s o t h a t root mean square values of d were used for calculating friction factors although t h e friction factor is clepenclrnt on tllr fifth power of the diameter.

I11 Appendix I the possible error in t h e value of obtained by this method is show11 t o he 0.21 per cell Measurement of Fluid Friction

Initially d a t a mere obtained for the pressurc loss due t o friction for t h e empty conduits. Only the d a t a for t h e sections betweell 20 a i d 30

ft.

from the air supply header were usecl in t h r analysis. T h i s cnsurecl t h a t entry effects wrre negligible.

I n Appendix I , the possible error in j is shown t o be approximately 2 per cent. a n d in Fig. 3 t h e deviation of the cxpcrimentnl results from the Moody2 "smooth pipe'' curve is less t h a n this possible error. T h u s for t h e range of Rcynoltls ~ i u n l b c r coverer1 by these tests t h e relativo r ~ u g h l i e s s of t h e concluit clitl not affect t h e pressure loss clue t o friction.

Finally t h e maill scries of tests consistc,d of ine:\suring the prcss~lrc loss due t o friction mhcn t h e contluits were partially iillctl. D a t : ~ were obtained for Rcyl~olcls i ~ ~ m b c r s

3. "1~'riction l7i~cto1.s for Pipe I"kllt"', hy L. I.". Moody, 7'rclrts- trcl~otis ASMI$ Vol. 66, 1944, p. 671

"

calculation it was assunled that the friction-factor- Reynolds-number relationshil:, for thc empty concluit was also applicable to the partially fillccl conduit when tho Reynolds number for this latter case was based upon the equivalent diameter of the partially filled conduit. This required t h a t the equivalent diameters be calcu1atc.d by an iterative method. The values of the friction factor used for these calculations were obtained from a large-scale plot o f f vs. Re which in turn mas bascd on data obtained from the same length of conduit when empty and when thr flow rate was measured with the same orifice plate. Tho accuracy of this calculation is discussed in Appendix

-

I

mhcrc it is shown t h a t values of d,/d have a possible error of 0.23 per cent. Figure 4 shows thc dimensionless ratio d,

/ d

plotted against Reynolds nuinbcr for the 1-in. conduit wit11 the various wire fills. The clata from thc other sizcs of conduit follow similar curvcs. Thcsc curves sllo\v t h a t for tlir rang? of Rcynolds numbers c~ovrrecl, t h r equivalent diameter is only slightly drprndcnt on Rrynolds numb:>r.

100" MANOME

UNION 8 PIPE PLUG PIEZOMETER RING

TE

TO VACU& PUMP

Fig. 2. Apparatus for i r ~ e a s u r i ~ ~ g the inside diameter of

c o ~ i d u i t .

Fig. 3. Friction f a c t o r 2 Reynolds number for the em- pty conduit compared with Moody 'sxnooth pipe' curve.

The values of

[/,/h

for a Rcy~iolds numbcr of 10,00 taken from Fig. 4 and thc other sirni1:~r

-

graphs arc sho in Fig. 5 plotted against the ratio d,/rl. I n this graph of the data fall along the three straight lines

_-

(1, /d = 1 - BN(d,/7) ₍₄₎

\vhc~rc

B1

= 0.407

B? = 0.762

B 3 = 1.050

The clata for thc tests on rubbcr-i~lsulsted wires and for wircs tnristcd thrrc turns in cac.11 10-ft. section fall on thc same curvcs in Fig. 5 as all the otllor clnta. This satis- factory correlation of thc rcsults suggests that all of the signific~ant variahlrs have been talien into account.

CONCLUSION

To ralculate thc pressure 1ossc.s duo to friction in a partially fillccl coiltluit it is only nrccssary to dcterminc a n equivalent cliametcr using thc appropriate in equation ( 4 ) , and then to calculate the pressure drop due to friction for this cquivaleilt conduit in thc usual may.

Thrsc results should be valid for all sizes of circular coiitluit containing circular cylinders as long as the ratio of fill diamctcr to conduit diameter does not exceed 0.3. Thc results arc limited to smooth conduits where entry effects are negligible, and t o Reynolds numbcrs in the range 5,000 to 50,000 but extrapolation to higher Reynolds numbers should not cause much error.

Acknowledgments

The author mishcs to thank Dr. N. 13. Hutcheon and Mr. il. G. Wilsoil for their helpful criticism of the paper; and i\I(,ssrs. J. 13erndt, P. Janecck and R . Evans for their assistance with the test work and calculations. This paper is a contribution froin the Ilivision of Building Research of the National Research Council of Canada and is pub- lished with t h r approval of the Director.

APPENDIX I. DISCUSSION OF ERRORS

( 1 ) Possible Error in the Mean Diameter - of t h e Conduit

conduit dia-

diameter to

able 11. Typical Data For Determining Pipe Friction Factor

(2) Possible Error in the Calculated Friction Factor The expression uscd to calculatc f was:

.32 f 0.05 mm. H 2 0 0.00 f 0.05 nlm. H 2 0

725

x

10-2)5 f t . 5

.OO 0.005

ft.

(3) Possible Error in Calculated Reynolds Nu

Thc expression used to calculate Re was:

where the quantities have the units given in Table I

Table 111 contains represcntative values for a 10- section of 1-in. conduit.

(4) Possible Error in de/Z

The expression used t o calculate d e was: